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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">b1</font> being  <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">b2</font> being  <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b3">b2</font> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Olive" title="b3">b2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">b</font> be   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b1</font> being  <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b2">b2</font> being  <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">B</font>  st <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b2">b2</font> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Olive" title="b1">b1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Olive" title="b2">b2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">b1</font> be   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b2</font> being  <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">B</font>  st <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">b2</font> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Olive" title="b1">b2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">b2</font> be   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c5">b2</font> implies  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c5">b2</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c5">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c5">b2</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c6">B1</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">b1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">B2</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c5">b2</font>;<br/>
<a NAME="E2:5"/><i><font color="Green" title="E7">A2a</font></i>: 
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_004.html#T1" target="_self" title="GLIB_004:th.1">Th1</a></i>;<br/>
<a NAME="E3:5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E4:5"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:5"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:5"/><b>then </b><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">b1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<b>by </b><i><a class="txt" href="#E6:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">B1</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">b1</font> as   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="#E2:5"><i><font color="Green" title="E7">A2a</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a>, <a class="ref" href="polynom1.html#D8" title="POLYNOM1:def.8">POLYNOM1:def 8</a></i>;<br/>
<a NAME="E9:5"/><i><font color="Green" title="E9">A4a</font></i>: 
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_004.html#T1" target="_self" title="GLIB_004:th.1">Th1</a></i>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c5">b2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">B</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">B2</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c5">b2</font> as   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="#E9:5"><i><font color="Green" title="E9">A4a</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a>, <a class="ref" href="polynom1.html#D8" title="POLYNOM1:def.8">POLYNOM1:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E13:5"/><i><font color="Green" title="E10">A5</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c10">f1</font>
 <b>and </b><br/><a NAME="E14:5"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D3" title="UPROOTS:def.3">UPROOTS:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E16:5"/><i><font color="Green" title="E12">A7</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c11">F1</font>
 <b>and </b><br/><a NAME="E17:5"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c11">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D3" title="UPROOTS:def.3">UPROOTS:def 3</a></i>;<br/>
<a NAME="E18:5"/><i><font color="Green" title="E14">A9</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E19:5"/><i><font color="Green" title="E15">A10</font></i>: 
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b1</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/>
<a NAME="E20:5"/><b>then </b><i><font color="Green" title="E16">A11</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E14:5"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="#E18:5"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E21:5"/><i><font color="Green" title="E17">A12</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E22:5"/>
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">B1</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/>
<a NAME="E23:5"/><b>then </b><i><font color="Green" title="E18">A13</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E17:5"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="#E21:5"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<div><i><font color="Green" title="E19">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font> implies <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font> ) &amp; ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:5_3_1"/><i><font color="Green" title="E20">A15</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font></span><br/><a NAME="E2:5_3_1"/><b>then </b><i><font color="Green" title="E21">A16</font></i>: 
<font color="Maroon" title="c4">b1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:5_3_1"/>
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/><a NAME="E4:5_3_1"/><b>then </b>
<font color="Maroon" title="c8">B1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="#E1:5_3_1"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E5:5_3_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font>
 <b>by </b><i><a class="txt" href="#E2:5_3_1"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:5_3"/><i><font color="Green" title="E20">A17</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font></span><br/><a NAME="E3:5_3"/><b>then </b><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c8">B1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_3_2"/>
(  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_3_2_1"><b>suppose </b></a><a NAME="E1:5_3_2_1"/>
not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font></span><br/><div class="add"><b>then </b><font color="Maroon" title="c8">B1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="polynom1.html#T56" title="POLYNOM1:th.56">POLYNOM1:56</a></i>
;<br/><b>hence </b><a NAME="E3:5_3_2_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font>
 <b>by </b><i><a class="txt" href="#E2:5_3"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_3_2_2"><b>suppose </b></a><a NAME="E1:5_3_2_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font></span><br/><div class="add"><a NAME="E2:5_3_2_2"/><b>then </b>
<font color="Maroon" title="c8">B1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E3:5_3_2_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font>
 <b>by </b><i><a class="txt" href="#E3:5_3"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E25:5"/><b>then </b><i><font color="Green" title="E20">A19</font></i>: 
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c8">B1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E26:5"/><i><font color="Green" title="E21">A20</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">F1</font>
 
<b>by </b><i><a class="txt" href="#E20:5"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E23:5"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="#E24:5"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">f1</font> implies <font color="Maroon" title="c10">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font> )</span><br/><b>assume </b><a NAME="E1:5_4"/><i><font color="Green" title="E22">A21</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span><br/><a NAME="E2:5_4"/><i><font color="Green" title="E23">A22</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E20:5"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E1:5_4"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>thus </b><font color="Maroon" title="c10">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font> = 
<font color="Maroon" title="c4">b1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
<b>by </b><i><a class="txt" href="#E14:5"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="#E20:5"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E1:5_4"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c8">B1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
<b>by </b><i><a class="txt" href="#E18:5"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="#E19:5"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="#E2:5_4"><i><font color="Green" title="E23">A22</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c11">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="txt" href="#E17:5"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="#E20:5"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E25:5"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="#E1:5_4"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E28:5"/><b>then </b><i><font color="Green" title="E22">A23</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c4">b1</font>
 
<b>by </b><i><a class="txt" href="#E13:5"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="#E16:5"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="#E26:5"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">f2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E30:5"/><i><font color="Green" title="E23">A24</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c5">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c12">f2</font>
 <b>and </b><br/><a NAME="E31:5"/><i><font color="Green" title="E24">A25</font></i>: 
<font color="Maroon" title="c12">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D3" title="UPROOTS:def.3">UPROOTS:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">F2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E33:5"/><i><font color="Green" title="E25">A26</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c9">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c13">F2</font>
 <b>and </b><br/><a NAME="E34:5"/><i><font color="Green" title="E26">A27</font></i>: 
<font color="Maroon" title="c13">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D3" title="UPROOTS:def.3">UPROOTS:def 3</a></i>;<br/>
<a NAME="E35:5"/><i><font color="Green" title="E27">A28</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E36:5"/><i><font color="Green" title="E28">A29</font></i>: 
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/>
<a NAME="E37:5"/><b>then </b><i><font color="Green" title="E29">A30</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E31:5"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="#E35:5"><i><font color="Green" title="E27">A28</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E38:5"/><i><font color="Green" title="E30">A31</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E39:5"/>
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">B2</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/>
<a NAME="E40:5"/><b>then </b><i><font color="Green" title="E31">A32</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E34:5"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="#E38:5"><i><font color="Green" title="E30">A31</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font> implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font> ) &amp; ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:5_5_1"/><i><font color="Green" title="E32">A33</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font></span><br/><a NAME="E2:5_5_1"/><b>then </b><i><font color="Green" title="E33">A34</font></i>: 
<font color="Maroon" title="c5">b2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:5_5_1"/>
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/><a NAME="E4:5_5_1"/><b>then </b>
<font color="Maroon" title="c9">B2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E1:5_5_1"><i><font color="Green" title="E32">A33</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E5:5_5_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font>
 <b>by </b><i><a class="txt" href="#E2:5_5_1"><i><font color="Green" title="E33">A34</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:5_5"/><i><font color="Green" title="E32">A35</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font></span><br/><a NAME="E3:5_5"/><b>then </b><i><font color="Green" title="E33">A36</font></i>: 
<font color="Maroon" title="c9">B2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_5_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_5_2"/>
(  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_5_2_1"><b>suppose </b></a><a NAME="E1:5_5_2_1"/>
not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font></span><br/><div class="add"><b>then </b><font color="Maroon" title="c9">B2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="polynom1.html#T56" title="POLYNOM1:th.56">POLYNOM1:56</a></i>
;<br/><b>hence </b><a NAME="E3:5_5_2_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font>
 <b>by </b><i><a class="txt" href="#E2:5_5"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_5_2_2"><b>suppose </b></a><a NAME="E1:5_5_2_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font></span><br/><div class="add"><a NAME="E2:5_5_2_2"/><b>then </b>
<font color="Maroon" title="c9">B2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E3:5_5_2_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font>
 <b>by </b><i><a class="txt" href="#E3:5_5"><i><font color="Green" title="E33">A36</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E42:5"/><b>then </b><i><font color="Green" title="E32">A37</font></i>: 
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c9">B2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">f2</font> implies <font color="Maroon" title="c12">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k</font> )</span><br/><b>assume </b><a NAME="E1:5_6"/><i><font color="Green" title="E33">A38</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k</font></span><br/><a NAME="E2:5_6"/><i><font color="Green" title="E34">A39</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E37:5"><i><font color="Green" title="E29">A30</font></i></a>, <a class="txt" href="#E1:5_6"><i><font color="Green" title="E33">A38</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>thus </b><font color="Maroon" title="c12">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k</font> = 
<font color="Maroon" title="c5">b2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font></span>)</span>
<b>by </b><i><a class="txt" href="#E31:5"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="#E37:5"><i><font color="Green" title="E29">A30</font></i></a>, <a class="txt" href="#E1:5_6"><i><font color="Green" title="E33">A38</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c9">B2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font></span>)</span>
<b>by </b><i><a class="txt" href="#E35:5"><i><font color="Green" title="E27">A28</font></i></a>, <a class="txt" href="#E36:5"><i><font color="Green" title="E28">A29</font></i></a>, <a class="txt" href="#E2:5_6"><i><font color="Green" title="E34">A39</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c13">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k</font>
<b>by </b><i><a class="txt" href="#E34:5"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="#E37:5"><i><font color="Green" title="E29">A30</font></i></a>, <a class="txt" href="#E42:5"><i><font color="Green" title="E32">A37</font></i></a>, <a class="txt" href="#E1:5_6"><i><font color="Green" title="E33">A38</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E44:5"/><b>then </b><i><font color="Green" title="E33">A40</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c9">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c5">b2</font>
 
<b>by </b><i><a class="txt" href="#E30:5"><i><font color="Green" title="E23">A24</font></i></a>, <a class="txt" href="#E33:5"><i><font color="Green" title="E25">A26</font></i></a>, <a class="txt" href="#E40:5"><i><font color="Green" title="E31">A32</font></i></a>, <a class="txt" href="#E42:5"><i><font color="Green" title="E32">A37</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a>, <a class="txt" href="#E31:5"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="#E35:5"><i><font color="Green" title="E27">A28</font></i></a>, <a class="txt" href="#E36:5"><i><font color="Green" title="E28">A29</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c9">B2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:5_7"/><i><font color="Green" title="E34">A41</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c9">B2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E2:5_7"/><i><font color="Green" title="E35">A42</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:5_7"/><i><font color="Green" title="E36">A43</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_7_1"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> or  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_7_1_1"><b>suppose </b></a><a NAME="E1:5_7_1_1"/><i><font color="Green" title="E37">A44</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c9">B2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:5_7_1_1"/><b>then </b><i><font color="Green" title="E38">A45</font></i>: 
not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font>
 <b>by </b><i><a class="txt" href="#E3:5_7"><i><font color="Green" title="E36">A43</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E3:5_7_1_1"/><i><font color="Green" title="E39">A46</font></i>: 
<font color="Maroon" title="c8">B1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E2:5_7"><i><font color="Green" title="E35">A42</font></i></a>, <a class="txt" href="#E1:5_7_1_1"><i><font color="Green" title="E37">A44</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><font color="Maroon" title="c9">B2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="txt" href="#E2:5_7_1_1"><i><font color="Green" title="E38">A45</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="polynom1.html#T56" title="POLYNOM1:th.56">POLYNOM1:56</a></i>
;<br/><b>hence </b><font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">B1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E2:5_7_1_1"><i><font color="Green" title="E38">A45</font></i></a>, <a class="txt" href="#E3:5_7_1_1"><i><font color="Green" title="E39">A46</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">B1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c9">B2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_7_1_2"><b>suppose </b></a><a NAME="E1:5_7_1_2"/><i><font color="Green" title="E37">A47</font></i>: 
not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c9">B2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>then </b><i><font color="Green" title="E38">A48</font></i>: <font color="Maroon" title="c8">B1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="txt" href="#E2:5_7"><i><font color="Green" title="E35">A42</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="polynom1.html#T56" title="POLYNOM1:th.56">POLYNOM1:56</a></i>
;<br/><a NAME="E3:5_7_1_2"/><i><font color="Green" title="E39">A49</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">b2</font>
 <b>by </b><i><a class="txt" href="#E1:5_7"><i><font color="Green" title="E34">A41</font></i></a>, <a class="txt" href="#E3:5_7"><i><font color="Green" title="E36">A43</font></i></a>, <a class="txt" href="#E1:5_7_1_2"><i><font color="Green" title="E37">A47</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:5_7_1_2"/><b>then </b>
<font color="Maroon" title="c9">B2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><b>hence </b><font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">B1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E2:5_7_1_2"><i><font color="Green" title="E38">A48</font></i></a>, <a class="txt" href="#E3:5_7_1_2"><i><font color="Green" title="E39">A49</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">B1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c9">B2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E46:5"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c5">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E28:5"><i><font color="Green" title="E22">A23</font></i></a>, <a class="txt" href="#E44:5"><i><font color="Green" title="E33">A40</font></i></a>, <a class="ref" href="uproots.html#T17" title="UPROOTS:th.17">UPROOTS:17</a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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